For \(X \times Y \in W^{s,p}(M) \times W^{s-1,p}(M)\) , \(\dim M =2\) , we can argue as in the proof of Theorem REF , because both local smoothing and Strichartz estimates remain true in the variable coefficient case. Indeed, variable-coefficient decoupling was proved by Beltran–Hickman–Sogge [1]} and local-in-time Strichartz estimates remain true on compact manifolds, as proved by Kapitanskii [2]}, [3]}. These are the key ingredients for the iteration argument in Section .
